any easy trick to solve

1. time period for two block and one spring system and also its max. and min. elongation. spring is in between the blocks

2. when a constant force is also applied on any one of the blocks

a. towards the another block

b. away from another block

To tackle the problem of a two-block and one-spring system, we need to break it down into manageable parts. Let's first consider the time period of the system and then explore the maximum and minimum elongation of the spring. After that, we can analyze the effects of applying a constant force on one of the blocks.

Understanding the System

Imagine two blocks, Block A and Block B, connected by a spring. The spring is placed between them, and both blocks can move freely. The system's behavior is influenced by the spring's properties and the masses of the blocks.

Time Period Calculation

The time period (T) of a mass-spring system can be derived from the formula:

• T = 2π√(m/k)

Where:

• m is the effective mass of the system.
• k is the spring constant.

For two blocks connected by a spring, the effective mass (m) can be calculated as:

• m = (m1 * m2) / (m1 + m2)

Here, m1 and m2 are the masses of Block A and Block B, respectively. Once you have the effective mass, you can substitute it into the time period formula to find T.

Maximum and Minimum Elongation

The maximum elongation occurs when the system is at its maximum displacement from the equilibrium position, while the minimum elongation is when the blocks are closest together. To find these values, consider the forces acting on the spring:

• Maximum elongation can be calculated when the system is displaced and released, leading to oscillations.
• Minimum elongation occurs when both blocks are at rest, and the spring is neither compressed nor stretched.

In a static scenario, if you know the initial displacement (x0), the maximum elongation would be x0 + the amplitude of oscillation, while the minimum elongation would be x0 - the amplitude.

Effect of a Constant Force

Now, let’s consider the scenario where a constant force is applied to one of the blocks.

Force Towards the Other Block

If a constant force is applied to Block A towards Block B, the spring will compress. The system will reach a new equilibrium position where the spring force balances the applied force. The elongation of the spring can be calculated using Hooke's Law:

• F = k * x

Where F is the applied force, k is the spring constant, and x is the elongation. Rearranging gives:

• x = F/k

This elongation will be the new maximum elongation of the spring when the force is applied.

Force Away from the Other Block

Conversely, if the force is applied away from Block B, the spring will stretch. Again, using Hooke's Law, the elongation can be calculated in the same manner:

• x = F/k

In this case, the spring will also reach a new equilibrium position, but the elongation will reflect the distance created by the force pulling the blocks apart.

Summary

In summary, the time period of the two-block and one-spring system can be calculated using the effective mass and spring constant. The maximum and minimum elongation depend on the initial displacement and the forces acting on the spring. When a constant force is applied, it alters the equilibrium position, leading to new elongation values based on the direction of the force. Understanding these principles allows you to analyze various scenarios in spring-mass systems effectively.